Default probabilities, credit derivatives, and computational issues.

Speaker: 

Professor Jean-Pierre Fouque

Institution: 

UCSB

Time: 

Tuesday, May 29, 2007 - 11:00am

Location: 

MSTB 254

The two main approaches to modeling defaults, structural and intensity based, will be reviewed. We show that perturbation methods are useful in approximating default probabilities in the context of stochastic volatility models. We then consider the case of many names and we discuss various ways of creating correlation of defaults. In highly-dimensional models, Monte Carlo simulations remain a powerful tool for computing prices of credit derivatives such as CDO's tranches and associated greeks. We propose an interactive particle system approach for computing the small probabilities involved in these financial instruments.

Diffusions in random environment and ballistic behavior.We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence.

Speaker: 

Professor Tom Schmitz

Institution: 

UCLA

Time: 

Tuesday, May 15, 2007 - 11:00am

Location: 

MSTB 254

We study diffusions in random environment in higher dimensions. We assume that the environment is stationary and obeys finite range dependence. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average over the environment. One then obtains the so-called annealed measures, that are typically non-Markovian measures.
Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime
('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce
conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix.
As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment.

Large deviations for partition functions of directed polymers and other models in an IID field.

Speaker: 

Professor Iddo Ben-Ari

Institution: 

UCI

Time: 

Tuesday, May 8, 2007 - 11:00am

Location: 

MSTB 254

Consider the partition function of a directed polymer in an IID
field. Under some mild assumptions on the field, it is a well-known fact
that the free energy of the polymer is equal to some deterministic constant
for almost every realization of the
field and that the upper tail large deviations is exponential. In this
talk I'll discuss the lower tail large deviations and present a method
for estimating it. As a consequence, I'll show that the lower
tail large deviations exhibits three regimes, determined by the
tail of the negative part of the field. The method applies to other
oriented models and can be adapted to non-oriented models as well. This
work extends the results of a recent paper by Cranston Gautier and
Mountford. A preprint is availabe on www.math.uci.edu/~ibenari

Asymptotic Enumeration of Spanning Trees via Traces and Random Walks

Speaker: 

Professor Russell Lyons

Institution: 

Indiana University

Time: 

Tuesday, May 22, 2007 - 11:00am

Location: 

MSTB 254

Methods of enumeration of spanning trees in a finite graph and relations to
various areas of mathematics and physics have been investigated for more
than 150 years. We will review the history and applications. Then we will
give new formulas for the asymptotics of the number of spanning trees of a
graph. A special case answers a question of McKay (1983) for regular
graphs. The general answer involves a quantity for infinite graphs that we
call ``tree entropy", which we show is a logarithm of a normalized
[Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs.
Proofs involve new traces and the theory of random walks.

Lyapounov norms for random walks in a random potential.

Speaker: 

Professor Nikolaos Zygouras

Institution: 

USC

Time: 

Tuesday, May 1, 2007 - 11:00am

Location: 

MSTB 254

Abstract : We consider a simple random walk on Z
, d > 3. We also consider
a collection of i.i.d. positive and bounded random variables ( V? (x) )x?Z d , which will
serve as a random potential. We study the annealed and quenched cost to perform
long crossings in the random potential ? + ? V? (x), where ? is positive constant
and ? > 0 small enough . These costs are measured by the Lyapounov norms We
prove the equality of the annealed and the quenched norm. We will also discuss the
relation between the Lyapounov norms and the path behavior of the random walk
in the random potential.

Don't love your children equally: the advantages of asymmetric damage segregation in fissioning organisms.

Speaker: 

Professor Steven N. Evans

Institution: 

U.C. Berkeley Statistics, Mathematics

Time: 

Tuesday, March 27, 2007 - 11:00am

Location: 

MSTB 254

A number of recent experiments have shown that several organisms
that reproduce by fissioning (e.g. E. coli bacteria)
don't share the cellular damage they have
accumulated during their lifetime equally among their offspring. Using
a stochastic PDE model, David Steinsaltz and I have shown that under quite
general conditions the optimal asymptotic growth rate for a population
of fissioning organisms is obtained when there is a non-zero but moderate
amount of preferential segregation of damage -- too much or too little
asymmetry is counter-productive. The proof uses some new results of ours
on quasi-stationary distributions of one-dimensional diffusions and
some Sturm-Liouville theory. The talk is intended for a probability
audience and I won't assume any knowledge of biology.

Don't love your children equally: the advantages of asymmetric damage segregation in fissioning organisms.

Speaker: 

Professor Steven Evans

Institution: 

U.C. Berkeley Statistics, Mathematics

Time: 

Monday, March 27, 2006 - 11:00pm

Location: 

MSTB 254

A number of recent experiments have shown that several organisms
that reproduce by fissioning (e.g. E. coli bacteria)
don't share the cellular damage they have
accumulated during their lifetime equally among their offspring. Using
a stochastic PDE model, David Steinsaltz and I have shown that under quite
general conditions the optimal asymptotic growth rate for a population
of fissioning organisms is obtained when there is a non-zero but moderate
amount of preferential segregation of damage -- too much or too little
asymmetry is counter-productive. The proof uses some new results of ours
on quasi-stationary distributions of one-dimensional diffusions and
some Sturm-Liouville theory. The talk is intended for a probability
audience and I won't assume any knowledge of biology.

Horseshoes in Multidimensional Scaling

Speaker: 

Professor Sharad Goel

Institution: 

USC

Time: 

Tuesday, March 20, 2007 - 11:00am

Location: 

MSTB 254

Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint proximities. I'll discuss MDS applied to a specific dataset: the 2005 United States House of Representatives roll call votes. In this case, MDS outputs 'horseshoes' that are characteristic of dimensionality reduction techniques. I'll show that in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide insight into manifold learning in the special case where the manifold is a curve. This work is joint with Persi Diaconis and Susan Holmes.

Pages

Subscribe to RSS - Combinatorics and Probability